Compensating for hysteresis

ABSTRACT

A method and apparatus for compensating for hysteresis in a system, the method comprising: determining a first estimate of a system parameter using the Preisach model; measuring a value of a system parameter; and determining an updated estimate of the estimated system parameter using the measured value of a system parameter. The measured system parameter may be the same system parameter as the system parameter for which the first estimate is determined using the Preisach model, or it may be a further system parameter that is a different system parameter to the system parameter for which the first estimate is determined using the Preisach model. Determining the updated estimate of the estimated system parameter may further use the first estimate of the estimated system parameter, or it may further use a corresponding determined value of the further system parameter.

FIELD OF THE INVENTION

The present invention relates to compensating for hysteresis, and otherprocesses related to hysteresis.

BACKGROUND

The exhibition of hysteresis by certain materials used in certainsystems, for example those used in a deformable mirror, is known.

A known problem relating to hysteresis in deformable mirrors is asfollows. An input voltage is applied to an actuator in a deformablemirror, which causes it to change shape. When the input voltage isturned off, the actuator will return to its original shape in time, butin a slightly different manner. It is likely that the actuator willexperience a second applied input voltage before it has returned to itsoriginal position. This causes the actuator to expand and contract in anunintended fashion. Thus, there is a degree of inaccuracy in thedeformable mirror system.

Known methods of reducing the degree of this inaccuracy (i.e. methods ofcompensating for hysteresis in, for example, a deformable mirror)implement the Preisach model to model hysteresis in the material, andthen implement an Inverse Preisach model to reduce the inaccuracy causedby hysteresis. Conventional applications of the Inverse Preisach modelrequire large amounts of processing, for example much greater amounts ofprocessing than is typically required for the forward Preisach model.

Conventionally, the Inverse Preisach model is implemented using a linearinterpolation based inversion algorithm, which requires large amounts ofprocessing. Also, an increasing input to the system tends not toconsistently lead to an increasing output from the system. This cancause interpolation problems.

The remainder of this section introduces Preisach model terminology usedlater below in the description of embodiments of the present invention.

The Preisach model describes hysteresis in terms of an infinite set ofelementary two-valued hysteresis operators (hysterons).

FIG. 1 is a schematic graph showing a typical input-output loop of asingle two-valued relay hysteron, referred to hereinafter as a“hysteron”. The x-axis of FIG. 1 represents an input voltage to thesystem, and is hereinafter referred to as the “input x”. For example,the input x is the input voltage applied to an actuator that deforms adeformable mirror. The y-axis of FIG. 1 represents an output voltagefrom the system, and is hereinafter referred to as the “output y”. Forexample, the output y is the displacement by which a deformable mirroris deformed by an actuator that has received an input voltage, e.g. theinput x. FIG. 1 shows that the input x ranges from minus two to two.Also, the output y takes a value of zero or one. The ranges for theinput x and the output y are merely exemplary, and may be differentappropriate values. The output level of one corresponds to the systembeing switched ‘on’, and the output level of zero corresponds to thesystem being switched ‘off’. The zero output level is shown in FIG. 1 asa bold line and is indicated by the reference numeral 10. The outputlevel of one is shown in FIG. 1 as a bold line and is indicated by thereference numeral 12. FIG. 1 shows an ascending threshold α at aposition corresponding to x=1, and a descending threshold β at aposition corresponding to x=−1.

Graphically, FIG. 1 shows that if x is less than the descendingthreshold β, i.e. −2<x<−1, the output y is equal to zero (off). As theinput x is increased from its lowest value (minus two), the output yremains at zero (off) until the input x reaches the ascending thresholdα, i.e. as x increases, y remains at 0 (off) if −2<x<1. At the ascendingthreshold α, the output y switches from zero (off) to one (on). Furtherincreasing the input x from one to two has no change of the output valuey, i.e. the hysteron remains switched ‘on’. As the input x is decreasedfrom its highest value (two), the output y remains at one (on) until theinput x reaches the descending threshold β, i.e. as x decreases, yremains at 1 if −1<x<2. At the descending threshold β, the output yswitches from one (on) to zero (off). Further decreasing the input xfrom minus one to minus two has no change of the output value y, i.e.the hysteron remains switched ‘off’.

Thus, the hysteron takes the path of a loop, and its subsequent statedepends on its previous state. Consequently, the current value of theoutput y of the complete hysteresis loop depends upon the history of theinput x.

Within a material, individual hysterons may have varied α and β values.The output y of the system at any instant will be equal to the sum overthe outputs of all of the hysterons. The output of a hysteron withparameters α and β is denoted as ξ_(αβ)(x). Thus, the output y of thesystem is equal to the integral of the outputs over all possiblehysteron pairs, i.e.

$y = {\underset{\alpha \geq \beta}{\int\int}{\mu \left( {\alpha,\beta} \right)}{\xi_{\alpha\beta}(x)}{\alpha}{\beta}}$

where μ(α,β) is a weighting, or density, function, known as the Preisachfunction.

This formula represents the Preisach model of hysteresis. The input tothe system corresponds to the input of the Preisach model (these inputscorrespond to x in the above equation). The output of the systemcorresponds to the output of the Preisach model (these outputscorrespond to y in the above equation).

FIG. 2 is a schematic graph showing all possible α-β pairs for thehysterons in a particular material. All α and β pairs lie in a triangle20 shown in FIG. 2. The triangle 20 is bounded by: the minimum of theinput x (minus two); the maximum of the input x (two); and the line α=βline (since α≧β).

Increasing the input x from its lowest amount (minus 2) to a value x=u₁provides that all of the hysterons with an α value less than the inputvalue of u₁ will be switched ‘on’. Thus, the triangle 20 of FIG. 2 isseparated into two regions. The first region contains all hysterons thatare switched ‘on’, i.e. the output y equals a value of one. The secondregion contains all hysterons that are switched ‘off’, i.e. the output yequals a value of zero. FIG. 3 is a schematic graph showing the regionof all possible α-β pairs, i.e. the triangle 20, divided into the abovedescribed two regions by increasing the input x from its lowest amountto a value x=u₁. The first region, i.e. the region that contains allhysterons that are switched ‘on’ is hereinafter referred to as the“on-region 22”. The second region, i.e. the region that contains allhysterons that are switched ‘off’ is hereinafter referred to as the“off-region 24”.

Decreasing the input x from the value x=u₁ to a value x=u₂ provides thatall of the hysterons with a β value greater than the input value of u₂will be switched ‘off’. Thus, the on-region 22 and the off-region 24 ofthe triangle 20 change as the input x is decreased from the value x=u₁to the value x=u₂. FIG. 4 is a schematic graph showing the regions 22,24 of the triangle 20 formed by decreasing the input x from the valuex=u₁ to the value x=u₂, after having previously increased the input x asdescribed above with reference to FIG. 3.

An increasing input can be thought of as a horizontal link that movesupwards on the graph shown in FIGS. 2-4. Similarly, a decreasing inputcan be thought of as a vertical link that moves towards the left on thegraph shown in FIGS. 2-4.

By alternately increasing and decreasing the input x, the triangle 20 isseparated in to two regions, the boundary between which has a number ofvertices. FIG. 5 is a schematic graph showing the regions 22, 24 of thetriangle 20 formed by alternately increasing and decreasing the input x.Alternately increasing and decreasing the input x produces a “staircase”shaped boundary between the on-region 22 and the off-region 24,hereinafter referred to as the “boundary 300”. The boundary 300 has fourvertices, referred to hereinafter as the “first x-vertex 30” (which hascoordinates (α₁,β₁)), the “second x-vertex 32” (which has coordinates(α₂,β₁)), the “third x-vertex 34” (which has coordinates (α₂,β₂)), andthe “fourth x-vertex 36” (which has coordinates (α₃,β₂)).

In FIG. 5 the final voltage change in the input x is a decreasingvoltage change (as indicated by the vertical line from the thirdx-vertex 34 to the line α=β). However, it is possible for the finalvoltage change in the input x to be an increasing voltage change. Thiscould be considered to be followed by a decreasing voltage change ofzero for convenience.

For the Preisach Model to represent a material's behaviours, thematerial has to have the following two properties: the material musthave the wiping-out property, which provides that certain increases anddecreases in the input x can remove or ‘wipe-out’ x-vertices; and thematerial must have the congruency property, which states that all minorhysteresis loops that are formed by the back-and-forth variation ofinputs between the same two extremum values are congruent.

The output y of the system is dependent upon the size and shape of theon-region 22. The on-region 22, in turn, is dependent upon thex-vertices 30, 32, 34, 36. Thus, as described in more detail laterbelow, the output y of the system can be determined using the x-vertices30, 32, 34, 36 of the boundary between the on-region 22 and theoff-region 24.

The output y of the system illustrated by FIG. 5 is:

$\begin{matrix}{y = {\underset{\alpha \geq \beta}{\int\int}{\mu \left( {\alpha,\beta} \right)}{\xi_{\alpha\beta}(x)}{\alpha}{\beta}}} \\{= {{\underset{{on}\text{-}{region}}{\int\int}{\mu \left( {\alpha,\beta} \right)}{\xi_{\alpha\beta}(x)}{\alpha}{\beta}} + {\underset{{off}\text{-}{region}}{\int\int}{\mu \left( {\alpha,\beta} \right)}{\xi_{\alpha\beta}(x)}{\alpha}{\beta}}}}\end{matrix}$

In the on-region 22, all hysterons are switched on, and thereforeξ_(αβ)(x)=1. Similarly, in the off-region 24 all hysterons are switchedoff, and therefore ξ_(αβ)(x)=0. Thus,

$\begin{matrix}{y = {{\underset{{on}\text{-}{region}}{\int\int}{{\mu \left( {\alpha,\beta} \right)} \cdot 1 \cdot {\alpha}}{\beta}} +}} \\{= {\underset{{off}\text{-}{region}}{\int\int}{{\mu \left( {\alpha,\beta} \right)} \cdot 0 \cdot {\alpha}}{\beta}}} \\{= {\underset{{on}\text{-}{region}}{\int\int}{\mu \left( {\alpha,\beta} \right)}{\alpha}{\beta}}}\end{matrix}$

By considering the x-vertices on the boundary 300, it can be shown thatthe integral can be estimated as follows:

$y = {\sum\limits_{k = 1}^{n}\left( {y_{\alpha_{k}\beta_{k}} - y_{\alpha_{k}\beta_{k - 1}}} \right)}$

where: y_(α) _(i) _(β) _(j) is the output y resulting from increasingthe input voltage x from the minimum to α_(i) and then decreasing it toβ_(j);

-   -   β₀ is the minimum saturation voltage, i.e. minus two; and

n is the number of vertical trapezia formed by the x-vertices on theboundary 300, i.e. n is therefore equal to

$\left\lfloor {\frac{1}{2} \times {number}\mspace{14mu} {of}\mspace{14mu} {vertices}} \right\rfloor.$

In practice, to calculate the above equation, values of y_(αβ) for anumber of points in the triangle 20 are generated. Typically, a value ofy_(αβ) for each α-β pairs in a grid of α-β pairs in the triangle 20 iscalculated. This is done by increasing the input x from its minimum(minus two) to α, and then decreasing it to β, and measuring the outputy of the system. For α-β pairs not on the grid, a value of y_(αβ) isfound using bilinear interpolation, or linear interpolation, using α-βpairs on the grid.

SUMMARY OF THE INVENTION

In a first aspect the present invention provides a method ofcompensating for hysteresis in a system, the method comprising:determining a first estimate of a system parameter using the Preisachmodel; measuring a value of a system parameter; and determining anupdated estimate of the estimated system parameter using the measuredvalue of a system parameter.

The measured system parameter may be the same system parameter as thesystem parameter for which the first estimate is determined using thePreisach model; and determining the updated estimate of the estimatedsystem parameter may further use the first estimate of the estimatedsystem parameter.

The measured system parameter may be a further system parameter that isa different system parameter to the system parameter for which the firstestimate is determined using the Preisach model; and determining theupdated estimate of the estimated system parameter may further use adetermined value of the further system parameter, the determined valueof the further system parameter being determined in correspondence tothe first estimate of the estimated system parameter.

The estimated system parameter may be an input of the system; anddetermining a first estimate of the system parameter may comprisecalculating the formula:

$x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}$

where x is the estimate of the input of the system; and x_(γ) _(i) _(δ)_(j) is the input of the system resulting from increasing an output ofthe system from a minimum output to a value of γ_(i) and then decreasingit to a value of δ_(j).

Calculating the formula

$x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}$

may comprise: determining a value of x_(γδ) for each γ-δ pair in adiscrete set of γ-δ pairs; and for all required values of x_(γδ) where γand δ are not in the discrete set, determining a value of x_(γδ) using aprocess of interpolation using values of x_(γδ) where γ and δ are in thediscrete set.

The step of determining the updated estimate of a system parameter maycomprise: if γ_(n) and δ_(n) are in the discrete set, adjusting thevalue of x_(γ) _(n) _(δ) _(n) by an amount equal to a constantmultiplied by the difference between the first estimate of a systemparameter and the measured value of a system parameter; and if γ_(n) andδ_(n) are not in the discrete set, adjusting the values of x_(γδ) foreach γ-δ pair surrounding (γ_(n), δ_(n)) by an amount equal to aconstant multiplied by the difference between the first estimate of asystem parameter and the measured value of a system parameter.

The constant may be equal to 0.005.

The step of determining the updated estimate of a system parameter maycomprise: if γ_(n) and δ_(n) are in the discrete set, adjusting thevalue of x_(γ) _(n) _(δ) _(n) by an amount equal to a constantmultiplied by the difference between the determined value of the furthersystem parameter and the measured value of a system parameter; and ifγ_(n) and δ_(n) are not in the discrete set, adjusting the values ofx_(γδ) for each γ-δ pair surrounding (γ_(n), δ_(n)) by an amount equalto a constant multiplied by the difference between the determined valueof the further system parameter and the measured value of a systemparameter.

The estimated system parameter may be an output of the system; anddetermining a first estimate of the system parameter may comprisecalculating the formula:

$y = {\sum\limits_{k = 1}^{n}\left( {y_{\alpha_{k}\beta_{k}} - y_{\alpha_{k}\beta_{k - 1}}} \right)}$

where y is the estimate of the output of the system; and y_(α) _(i) _(β)_(j) is the output of the system resulting from increasing the input ofthe system from the minimum input to a value of α_(i) and thendecreasing it to a value of β_(j).

Calculating the formula

$y = {\sum\limits_{k = 1}^{n}\left( {y_{\alpha_{k}\beta_{k}} - y_{\alpha_{k}\beta_{k - 1}}} \right)}$

may comprise: determining a value of y_(αβ) for each α-β pair in adiscrete set of α-β pairs; and for all required values of y_(αβ) where αand β are not in the discrete set, determining a value of y_(αβ) using aprocess of interpolation using values of y_(αβ) where α and β flare inthe discrete set.

The step of determining the updated estimate of a system parameter maycomprise: if α_(n) and β_(n) are in the discrete set, adjusting thevalue of y_(α) _(n) _(β) _(n) by an amount equal to a constantmultiplied by the difference between the first estimate of a systemparameter and the measured value of a system parameter; and if α_(n) andβ_(n) are not in the discrete set, adjusting the values of y_(αβ) foreach α-β pair surrounding (α_(n), β_(n)) by an amount equal to aconstant multiplied by the difference between the first estimate of asystem parameter and the measured value of a system parameter.

The system may be an adaptive optics system.

In a further aspect the present invention provides apparatus adapted toperform the method of any of any of the above aspects.

In a further aspect the present invention provides a computer program orplurality of computer programs arranged such that when executed by acomputer system it/they cause the computer system to operate inaccordance with the method of any of the above aspects.

In a further aspect the present invention provides a machine readablestorage medium storing a computer program or at least one of theplurality of computer programs according to the above aspect.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic graph showing a typical input-output loop of asingle two-valued relay hysteron;

FIG. 2 is a schematic graph showing all possible α-β pairs for thehysterons in a particular material;

FIG. 3 is a schematic graph showing the region of all possible α-βpairs;

FIG. 4 is a schematic graph showing the on-region and off-region of theregion of all possible α-β pairs formed by decreasing the input x fromthe value x=u₁ to the value x=u₂;

FIG. 5 is a schematic graph showing the on-region and off-region of theregion of all possible α-β pairs formed by alternately increasing anddecreasing the input x;

FIG. 6 is a schematic illustration of a mirror interrogation system;

FIG. 7 is a schematic graph showing the space of possible increases anddecreases in the output y generated by alternately increasing anddecreasing the input x;

FIG. 8 is a process flow chart showing a method of implementing anInverse Preisach model;

FIG. 9 is a schematic graph showing a grid of γ-δ pairs defined on thespace of all possible γ-δ pairs; and

FIG. 10 is a schematic illustration of an adaptive optics system.

DETAILED DESCRIPTION

FIG. 6 is a schematic illustration of a mirror interrogation system 99in which an example Inverse Preisach model is performed. An adaptiveupdating process is used to adaptively update the Inverse Preisach modelin an embodiment of compensating for hysteresis, as described laterbelow with reference to FIG. 10, after the description of the InversePreisach model.

The mirror interrogation system 99 comprises a controller 100, adeformable mirror 101, a beam-splitter 106, and a wave-front sensor 108.The deformable mirror 101 comprises an actuator 102 and a mirror 104.

The controller 100 comprises an output and an input. The output of thecontroller 100 is connected to the actuator 102. The input of thecontroller is connected to the wave-front sensor 108. The controller 100receives a signal from the wave-front sensor 108. The controllerprocesses the signal received from the wave-front sensor 108, asdescribed in more detail later below. The controller 100 sends a controlsignal to the actuator 102. The control signal depends on the signalreceived by the controller 100 from the wave-front sensor 108, asdescribed in more detail later below.

The actuator 102 comprises an output and an input. The input of theactuator 102 is connected to the controller 100. The output of theactuator 102 is connected to the mirror 104. The actuator 102 receivesthe control signal from the controller 100. The actuator changes theshape of, or deforms, the mirror 104 via the actuator output, dependingon the received control signal. In this example, the control signalreceived by the actuator is an input voltage. This input voltage to theactuator corresponds to the input x for a Preisach model, as describedabove and referred to later below. The value of the input x determinesthe amount by which the actuator 102 deforms the mirror 104.

The mirror 104 is deformed by the actuator 102. The displacement of themirror, i.e. the amount by which the mirror 104 is deformed by theactuator 102, corresponds to the output y for a Preisach model, asdescribed above and referred to later below.

In this example, a beam of light is reflected and split by the mirrorinterrogation system 99. The beam of light is indicated by arrows inFIG. 6. For clarity and ease of understanding, the beam of light isshown as separate sections: a first section of the beam of light that isincident on the mirror 104, (the first section is hereinafter referredto as the “incident beam 110”); a second section of the beam of lightthat is reflected from the mirror 104 and is incident on thebeam-splitter 106 (the second is hereinafter referred to as the“reflected beam 112”); and a third section and a fourth section of thebeam of light that are formed by the beam splitter 106 splitting thereflected beam (the third section is hereinafter referred to as the“image beam 114” and the fourth section is hereinafter referred to asthe “feedback beam 116”).

The beam splitter splits the reflected beam 106 into the image beam 114and the feedback beam 116.

The feedback beam 116 is incident on the wave-front sensor 108.

The wave-front sensor 108 detects the feedback beam 116. The wave-frontsensor 108 measures a value of the curvature of the mirror 104. Thewave-front sensor 108 comprises an output. The wave-front sensor 108sends a signal corresponding to the detected feedback beam 116, i.e. asignal corresponding to the curvature of the mirror 104, to thecontroller 100 via the output of the wave-front sensor 108.

The controller 100 receives the signal corresponding to the detectedfeedback beam 116 from the wave-front sensor 108. In this example, thecontroller 100 comprises a processor (not shown). The processor uses thesignal corresponding to the detected feedback beam 116 to determine avalue corresponding to the displacement of the mirror 104, i.e. a valuefor the output y of the Preisach model. The processor further generatesa control signal, i.e. the input x for the Preisach model, using thedetermined output y. The control signal is sent to the actuator 102, andthe actuator 102 deforms the mirror 104 depending on the receivedcontrol signal as described above.

An example of performing the Inverse Preisach model in the mirrorinterrogation system 99 described above with reference to FIG. 6, willnow be described. In this example, a problem of obtaining an inverseresult for the Preisach model is addressed making use of aspects of theforward Preisach model, as opposed to using the conventional approach ofinterpolation-based techniques.

In this example, the deformable mirror 101 has the wiping-out propertyas described above for the forward Preisach model, and described belowfor this example.

As described in more detail above, alternately increasing and decreasingthe input x (which in this example is the input voltage applied to theactuator 102) produces the α-β graph shown in FIG. 5. The α-β graphshown in FIG. 5 shows the first x-vertex 30, the second x-vertex 32, thethird x-vertex 34, and the fourth x-vertex 36.

The output y (which in this example is the displacement of the mirror104 by the actuator 102) of the system is, in general, an increasingfunction with respect to the input x, i.e. as the input x increases, theoutput y increases, and as the input x decreases, the output ydecreases. Thus, alternately increasing and decreasing the input x,produces alternating increases and decreases in the output y. This isshown schematically in FIG. 7.

FIG. 7 is a schematic graph showing the space of possible increases anddecreases in the output y generated by the increasing and decreasing ofthe input x. The vertical axis of FIG. 7 is the level to which theoutput y is increased as the input x is increased, indicated by thereference sign γ. The horizontal axis of FIG. 7 is the level to whichthe output y is decreased as the input x is decreased, indicated by thereference sign δ. The space of all possible γ-δ pairs is indicated inFIG. 7 by the reference number 21. The space of γ-δ pairs 21 correspondsto the triangle 20 of all possible α-β pairs for the hysterons in aparticular material, as described above with reference to FIGS. 2-5. Thespace of γ-δ pairs 21 is bounded by: the minimum of the output y (zero);the maximum of the output (one); and the line γ=δ line (since γ≧δ).

FIG. 7 further shows the output y that is produced by the input x beingalternately increased and decreased according to the α-β graph shown inFIG. 5. In this example, the output y comprises four vertices,hereinafter referred to as the “first y-vertex 40” (which hascoordinates (γ₁, δ₁)), the “second y-vertex 42” (which has coordinates(γ₂, δ₁)), the “third y-vertex 44” (which has coordinates (γ₂, δ₂)), andthe “fourth y-vertex 46” (which has coordinates (γ₃, δ₂)).

The first x-vertex 30 of the boundary 300 shown in the α-β graph for theinput x (FIG. 5) corresponds to the first y-vertex 40 for the output yshown in FIG. 7. In other words, as the input x is increased to α₁, theoutput y correspondingly increases to γ₁. Then, as the input x isdecreased to β₁, the output y correspondingly decreases to δ₁.

The second x-vertex 32 of the boundary 300 shown in the α-β graph forthe input x (FIG. 5) corresponds to the second y-vertex 42 for theoutput y shown in FIG. 7. In other words, as the input x is decreased toβ₁, the output y correspondingly decreases to δ₁. Then, as the input xis increased to α₂, the output y correspondingly increases to γ₂.

The third x-vertex 34 of the boundary 300 shown in the α-β graph for theinput x (FIG. 5) corresponds to the third y-vertex 44 for the output yshown in FIG. 7. In other words, as the input x is increased to α₂, theoutput y correspondingly increases to γ₂. Then, as the input x isdecreased to β₂, the output y correspondingly decreases to δ₂.

The fourth x-vertex 36 of the boundary 300 shown in the α-β graph forthe input x (FIG. 5) corresponds to the fourth y-vertex 46 for theoutput y shown in FIG. 7. In other words, as the input x is decreased toβ₂, the output y correspondingly decreases to δ₂. Then, as the input xis increased to α₃, the output y correspondingly increases to γ₃.

In this example, the wiping-out property, which holds for the α-β graphfor the input x shown in FIG. 5, also holds for the γ-δ graph for theoutput y shown in FIG. 7. For example, if the input x is decreased fromits value at the fourth x-vertex 36 to the β-value value of the secondx-vertex 32, i.e. to β₁, then the second, third and fourth x-vertices32, 34, 36 are wiped out. Thus, the output y is only dependent on thefirst x-vertex 30. The first x-vertex 30 corresponds to the firsty-vertex 40 in FIG. 7, and so the output y depends only on the firsty-vertex 40, i.e. the second, third, and fourth y-vertices 42, 44, 46have been wiped out. A similar argument can be constructed for anincreasing value of the input x. In this example, the wiping-outproperty holds if y is an increasing function with respect to x. Inpractice this requirement can be assumed to hold (even though nomaterial is perfect and therefore y is not quite an increasing function)because the material properties are such that useful results arenevertheless produced.

In this example, the congruency property, which holds for the α-β graphfor the input x shown in FIG. 5, does not necessarily hold true for theinverse. For example, the output y is to be increased from y₁ to y₂. Todo this, the input x is increased from x₁ to x₂. To return the output yto y_(i), the input x is decreased to x₁. Thus, each time the output yis increased from y₁ to y₂ and decreased to y₁ again, an identical‘loop’ of input values is created. However, the input x (and/or theoutput y) may have a history that requires a different value of x₁ tobring the output to y₁, and a different value of x₂ to bring the outputto y₂. x₂ and x₁ are not necessarily the same, thus increasing theoutput y from y₁ to y₂ and back to y₁ may not result in congruent‘loops’ of the input x values for all input histories. In other words,the range over which the input x is varied to produce a particular loopin the output y changes with different input histories.

In this example, it is not necessary for the congruency property to holdfor the inverse because, in practice, the material tends not to have aperfect congruency property for the forward Preisach model. Also, minorhysteresis loops near the centre of the major hysteresis loop are likelyto all be very similar. In practice, the better the congruency propertyholds, the better the inverse Preisach model, herein described, willwork. In other words, the material does not have a perfect ‘forward’congruency property, so the ‘inverse’ congruency property tends not bedetrimentally limited. Also, for readings in the middle of thehysteresis loop, all of the loops are quite similar. Thus, thecongruency property tends to hold reasonably well in this middle range.

The formula for the inverse model is:

$x = {\underset{{\gamma \geq \delta}\mspace{14mu}}{\int\int}{\lambda \left( {\gamma,\delta} \right)}{ɛ_{\gamma\delta}(y)}{\gamma}{\delta}}$

Where: λ(γ,δ) is a density function; and

-   -   ε_(γδ)(y) is the output of an imaginary hysteron having        parameters γ and δ.

This formula represents an Inverse Preisach model of hysteresis for thepresent example. The input to the system, i.e. the input voltage to theactuator 102, corresponds to the output of the Inverse Preisach model(these correspond to x in the above equation). The output of the system,i.e. the displacement of the mirror 104, corresponds to the input of theInverse Preisach model (these correspond to y in the above equation).

This formula can be rewritten as a summation, in the same way as for theforward model:

$x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}$

where:

-   -   x_(γ) _(i) _(δ) _(j) is the input x resulting from increasing        the output y from the minimum (in this example, the minimum        output is zero) to γ_(i) and then decreasing it to δ_(j); and    -   δ₀ is the minimum output, i.e. zero.    -   n is the number of vertical trapezia formed by the y-vertices in        triangle 21, and is therefore equal to [½×number of vertices].

As described above for the forward Preisach model, values of x_(γδ)where γ and/or δ are not on the grid, is found using bilinear or linearinterpolation using determined values of x_(γδ) where γ and δ are on thegrid.

In this example, the minimum output is 0. However, in other examples,the minimum output is not zero. In examples in which the minimum outputis not zero, an appropriate additional constant is added to thesum/integral in the above equations for the inverse Preisach model toaccount for the non-zero minimum output.

In a corresponding way to performing the forward Preisach model, a valueof the input x is generated for each γ-δ pair in a grid of γ-δ. Thesevalues are hereinafter denoted as x_(γδ). For γ-δ pairs not on the grid,a value of x is found using bilinear interpolation, or linearinterpolation, using γ-δ pairs on the grid.

In this example, the grid input values are generated using the forwardPreisach model. The input x is increased slowly (i.e. in smallincrements) until y=γ, and then the input x is slowly decreased untily=δ. This example of implementing an Inverse Preisach model is describedin more detail later below, with reference to FIG. 7. The forwardPreisach model, utilised as described above, advantageously tends not tosuffer from creep. Also, the forward Preisach model advantageously tendsto be time-independent, and have perfect wiping-out and congruencyproperties. In other examples, the grid of input values x_(γδ) could begenerated by implementing the above method using the deformable mirrorinstead of the forward Preisach Model. However, the deformable mirrormay suffer from creep and may not perfectly satisfy the Preisachcriteria, i.e. the deformable mirror may not have perfect wiping-out andcongruency properties.

FIG. 8 is a process flow chart showing a method of implementing anInverse Preisach model according to the above described example.

At step s2, a grid of γ-δ pairs is defined on the space of all possibleγ-δ pairs. In other words, a grid of points is defined on the space ofγ-δ pairs 21 shown in FIG. 7.

FIG. 9 is a schematic graph showing a grid of γ-δ pairs (indicated bydots in the space of γ-δ pairs 21) defined on the space of all possibleγ-δ pairs (the space of γ-δ pairs 21).

At step s4, for a particular γ-δ pair, the input x to the system isslowly increased from the minimum input (minus two) until the output yof the system equals the value of y of the particular γ-δ pair.

At step s6, for the particular γ-δ pair, the input x to the system isslowly decreased until the output y of the system equals the value of δof the particular γ-δ pair.

At step s8, the value of the input x after having performed the steps s4and s6 above is stored in a table for that particular γ-δ pair, i.e. thevalue of the input x is stored as x_(γδ), as described above.

At step s10, the steps s4, s6 and s8 are repeated for all remaining γ-δpairs. Thus, for each value of the output y that is defined as a pointon the grid of γ-δ pairs in FIG. 8, a corresponding value x_(γδ) of theinput x that is required to produce such an output y, is determined andstored.

In this example, steps s2-s10, as described above, are performed onceand the grid of input values x_(γδ) is stored and used as a referencefor performing step s12.

At step s12, the input x for a series system of outputs is calculatedusing the formula (described earlier above):

$x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}$

where:

-   -   x_(γ) _(i) _(δ) _(j) is the input x resulting from increasing        the output y from the minimum to γ_(i) and then decreasing it to        δ_(j) (zero);    -   δ₀ is the minimum output, i.e. zero; and    -   n is the number of vertical trapezia formed by the y-vertices in        triangle 21, and is therefore equal to

$\left\lfloor {\frac{1}{2} \times {number}\mspace{14mu} {of}\mspace{14mu} {vertices}} \right\rfloor.$

A value of x_(γδ) for values of γ and/or δ not on the grid, is foundusing bilinear interpolation, or linear interpolation, using determinedgrid values of x_(γδ).

Thus, a method of performing an Inverse Preisach model is provided. Theprovided inverse procedure tends to be faster than conventional,iterative inverse procedures.

A further advantage is that algorithms prepared for the forward Preisachmodel tend to be usable (with the alternative grid of points) forimplementing the provided inverse procedure.

A further advantage of above described Inverse Preisach model is thatthe grid of γ-δ pairs provided by the process can be updated usingfeedback from sensors, for example the wave-front sensor 108. This tendsto allow for faster processing and more accurate estimations.

The above described Inverse Preisach model is used to compensate forhysteresis in a system. In the following example, the Inverse Preisachmodel is used to compensate for hysteresis in an adaptive optics system.

The adaptive optics system described in the following example comprisesthe same deformable mirror 101 (i.e. the same actuator 102 and mirror104) present in the mirror interrogation system 99. This is so that theinverse Preisach information generated by the mirror interrogationsystem 99 (as described above) can be used to compensate for hysteresisin the adaptive optics system, i.e. the inverse Preisach information isderived from the particular deformable mirror 101. Alternatively, inother examples a deformable mirror of the same type as the deformablemirror that has been interrogated and had inverse Preisach informationgenerated for it, for example a deformable mirror manufactured to thesame specifications as those of the interrogated deformable mirror, isused. In other examples, a deformable mirror is arranged as part of amirror interrogation system and as part of an adaptive optics system atthe same time.

FIG. 10 is a schematic illustration of an adaptive optics system 150 inwhich an example of compensating for hysteresis is implemented.

The adaptive optics system 150 comprises the controller 100, thedeformable mirror 101, a further beam splitter 152, and a furtherwave-front sensor 154. The deformable mirror 101 comprises the actuator102 and the mirror 104.

In this example, a beam of light is split and reflected by the adaptiveoptics system 150. The beam of light is indicated by arrows in FIG. 10.For clarity and ease of understanding, the beam of light is shown asseparate sections: a first section of the beam of light that is incidenton the further beam splitter 152, (the first section is hereinafterreferred to as the “further incident beam 156”); a second and a thirdsection of the beam of light that are formed by the further beamsplitter 152 splitting the further incident beam 156. The second sectionis hereinafter referred to as the “mirror beam 157”, and is reflected bythe mirror 104. The third section is hereinafter referred to as the“sensor beam 158”) and is incident on the further wave-front sensor 154.

In this example, the actuator 102 deforms the mirror 104 to generate aspherical surface to enable the system to correct for sphericalaberrations.

The further wave-front sensor 154 detects the sensor beam 158. Thefurther wave-front sensor 154 sends a signal corresponding to thedetected sensor beam 158, i.e. a signal corresponding to the furtherincident beam 156, to the controller 100.

The controller 100 receives the signal corresponding to the detectedsensor beam 158 from the further wave-front sensor 154. The processor inthe controller uses the signal corresponding to the detected sensor beam158 to determine a value corresponding to the displacement of the mirror104, i.e. a value for the output y of the Preisach model. The processorfurther generates a control signal using the above described InversePreisach model, i.e. the input x using the determined output y. Thecontrol signal is sent to the actuator 102, and the actuator 102 deformsthe mirror 104 depending on the received control signal. In this way,the hysteresis experienced by the mirror 104 resulting from operation inresponse to the further incident beam 156 is compensated for.

In the above examples, the light can be of any wavelength, for exampleinfra-red.

In the above described example, a grid of γ-δ pairs is defined. In thisexample, the spacing between the grid pairs is small. In the abovedescribed example 8001 grid points are implemented. Typically, thelarger the number of grid points, the more accurate the estimates of theinput x to the system are.

In the above examples, the output y (shown in FIG. 7) produced when theinput x is alternately increased and decreased according to the α-βgraph shown in FIG. 5, comprises four vertices. However, in otherexamples, the input x is alternately increased and decreased in adifferent manner, i.e. the α-β graph for input x has a different numberof vertices. Thus, in other examples, the output y produced by the inputx is different, for example, the output y may have a different number ofvertices in the γ-δ space.

In the above examples, the congruency property does not hold for theoutput y. However, in other examples, the congruency property does holdfor the output y.

In the above examples, the grid of γ-δ pairs, as shown in FIG. 9, is arectangular grid. However, in other examples the points of the γ-δ grid,i.e. the γ-δ pairs, are distributed in a different appropriate manner.For example, in other examples the grid of γ-δ pairs is a triangulargrid. The process of bilinear or linear interpolation used to determinevalues of x_(γδ) for values of γ and/or δ not on the grid, is modifiedaccordingly, or a different appropriate process is used, to account forthe grid of γ-δ pairs.

In the above described examples, the values of x_(γδ) for each γ-δ pairon the grid of γ-δ pairs are determined one at a time, as describedabove with reference to steps s4-s10 of the above described method, andFIG. 8. However, in other examples some or all of the values of x_(γδ)for each γ-δ pair on the grid of γ-δ pairs are determined concurrently,for example by implementing steps s4-s8 of the above described methodconcurrently on different specimens of a particular material.

In the above examples, the values of x_(γδ) for each γ-δ pair on thegrid of γ-δ pairs are determined by increasing the output to γ, thendecreasing the output to δ. However, in other examples, the values ofx_(γδ) are determined in a different manner. For example, a series ofx_(γδ) values could be found by increasing the output to a particularvalue, and then decreasing the output to a series of values, each valuelower than the last.

Returning to FIG. 6, the controller 100 implements inter alia thevarious method steps described above. The controller may be implementedor provided by configuring or adapting any suitable apparatus, forexample one or more computers or other processing apparatus orprocessors, and/or providing additional modules. The apparatus maycomprise a computer, a network of computers, or one or more processors,for implementing instructions and using data, including instructions anddata in the form of a computer program or plurality of computer programsstored in or on a machine readable storage medium such as computermemory, a computer disk, ROM, PROM etc., or any combination of these orother storage media.

An embodiment of an adaptive updating process, in which the abovedescribed example (or any variant thereof as described above) of anInverse Preisach model performed and adaptively updated, will now bedescribed. As part of this embodiment, the above described example of anInverse Preisach model (or any variant thereof as described above) isperformed as described above, except for where stated otherwise below.In this embodiment, the deformable mirror is arranged as part of themirror interrogation system and as part of the adaptive optics system atthe same time.

The above described formula used to determine the input x for a seriessystem of outputs is adapted as follows:

$\begin{matrix}{x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}} \\{= {\left\lbrack {{\sum\limits_{k = 1}^{n - 1}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)} - x_{\gamma_{n}\delta_{n - 1}}} \right\rbrack + x_{\gamma_{n}\delta_{n}}}}\end{matrix}$

where x is the estimation of the input to the system based on the abovedescribed embodiment of an Inverse Preisach model. Thus, the followingformula holds:

$x_{A} = {\left\lbrack {{\sum\limits_{k = 1}^{n - 1}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)} - x_{\gamma_{n}\delta_{n - 1}}} \right\rbrack + \left\lbrack {x_{\gamma_{n}\delta_{n}} + \left( {x_{A} - x} \right)} \right\rbrack}$

where x_(A) is the actual input to the system (required to produce thedesired output).

Thus, adjusting the value of x_(γ) _(n) _(δ) _(n) by an amountproportional to (x_(A)−x) increases the accuracy of the estimation ofthe input to the system x based on the above described example of anInverse Preisach model. If the γ-δ pair (γ_(n), δ_(n)) is a grid point,then x_(γ) _(n) _(δ) _(n) is adjusted by an amount proportional to(x_(A)−x). However, in this embodiment, the γ-δ pair (γ_(n), δ_(n)) isunlikely to be a grid point on the above described grid of γ-δ pairs.Therefore, the grid points surrounding the γ-δ pair (γ_(n), δ_(n)) areeach adjusted by an amount equal to k(x_(A)−x), where k is a constant.In this embodiment k=0.005. The value k=0.005 tends to provide the mostimproved estimate of the input x for the above described example of theInverse Preisach model. The value of x_(γ) _(n) _(δ) _(n) is calculatedas described above using bilinear or linear interpolation.

In practice, it may not be possible to determine a value of x_(A).However, the output y as a function of the input x tends to be smooth.Thus, the value of k(x_(A)−x) may be estimated by the following term:

K(y_(A)−δ_(n))

where:

K is a constant; and

y_(A) is the actual output of the system (after applying voltage x); and

δ_(n) is the required output.

The actual output of the system y_(A) may be determined by anyappropriate means. For example, the actual output of the system y_(A)may be determined by directly coupling a strain gauge or a capacitivesensor to the mirror 104, or by implementing the mirror interrogationsystem 99, as described above with reference to FIG. 6, i.e. by using awave-front sensor.

In the above embodiments, an adaptive updating process is used toadaptively update the Inverse Preisach model. However, in otherembodiments the adaptive updating process is used to update otherappropriate processes. For example, the adaptive updating process can beused to update the forward Preisach model, i.e. a further embodiment isprovided by implementing the above described forward Preisach model withthe above described adaptive updating process, as follows. The equationfor the actual output of the system is as follows (N.B. this correspondsto the earlier given equation for the actual input to the system in theprevious embodiment):

$y_{A} = {\left\lbrack {{\sum\limits_{k = 1}^{n - 1}\left( {y_{\alpha_{k}\beta_{k}} - y_{\alpha_{k}\beta_{k - 1}}} \right)} - y_{\alpha_{n}\beta_{n - 1}}} \right\rbrack + \left\lbrack {y_{\alpha_{n}\beta_{n}} + \left( {y_{A} - y} \right)} \right\rbrack}$

where y_(A) is the actual output of the system, and is the estimation ofthe output of the system based on the forward Preisach model. In thisembodiment, the value of y_(α) _(n) _(β) _(n) is adjusted by an amountproportional to (y_(A)−y) to increase the accuracy of the estimation ofthe output of the system y based on the above described forward Preisachmodel. If the α-β pair (α_(n), β_(n)) is a grid point, then y_(α) _(n)_(β) _(n) is adjusted by an amount proportional to (y_(A)−y). However,in this embodiment, the α-β pair (α_(n), β_(n)) is not a grid point onthe above described grid of α-β pairs. Therefore, the grid pointssurrounding the α-β pair (α_(n), β_(n)) are each adjusted by an amountequal to k(y_(A)−y), where k is a constant.

In the above embodiments, the value of the constant k used in theadaptive updating process is 0.005. However, in other embodimentsdifferent values of k are used. For example, in embodiments in which theadaptive updating process is used to adaptively update the forwardPreisach model, the value k=0.05 may be used. The value k=0.05 tends toprovide a particularly improved estimate of the output y for the forwardPreisach model.

In the above embodiments, the grid points surrounding the γ-δ pair(γ_(n), δ_(n)) are each adjusted by an amount equal to k(x_(A)−x) (orK(y_(A)−δ_(n))). For example, the four points surrounding the particulargrid point (γ_(n), δ_(n)) are each adjusted. In other examples adifferent number of points can be adjusted. For example, in otherembodiments more than four points surrounding the particular grid point(γ_(n), δ_(n)) are each adjusted by different amounts. An advantageprovided by this is that more accurate estimations tend to be produced.

The above described adaptive updating process tends to provide that agrid of γ-δ pairs in which the spacing between the grid pairs is small,is not necessary. Indeed, the adaptive updating process tends to providemore accurate results using a grid of γ-δ pairs in which the spacingbetween the grid pairs is larger. In this embodiment, 351 γ-δ pairs areutilised. This tends to advantageously allow for faster computation ofthe estimated values.

In the above embodiments, the value of the constant k used in theadaptive updating process is 0.005. However, in other embodimentsdifferent values of k are used.

The above described adaptive updating process advantageously providesthat the grid of γ-δ pairs (or the grid of α-β pairs) used in theprocess can be updated using feedback from sensors. This tends toprovide more accurate estimations.

In the above embodiments, the controller comprises a processor whichuses the signal corresponding to the detected feedback beam to determinea value corresponding to the displacement of the deformable mirror, i.e.a value for the output y of the Preisach model. However, in otherembodiments the value corresponding to the displacement of thedeformable mirror is determined in a different appropriate way. Forexample, in other embodiments the processor determines the displacementof the deformable mirror using signals from sensors that directlymeasure the displacement of the deformable mirror.

In the above embodiments, the deformable mirror comprises a mirror thatis deformed by an actuator. However, in other embodiments the deformablemirror is a different appropriate type of deformable mirror, for examplea bimorph mirror.

In a further embodiment, the actuator 102 is used to control theposition of the mirror 104 to generate a piston action, correctingphase. In a further embodiment, a discrete array of such phasecorrectors is used to generate a multi-element deformable mirror. Eachdiscrete corrector can be controlled as described above.

In the above embodiments, the output y of the Preisach model is a valueof the displacement of the deformable mirror. However, in otherembodiments the output y of the Preisach model is a differentappropriate parameter. For example, in other embodiments the output isthe measured value of the feedback beam detected by the wave-frontsensor.

In the above embodiments, the input x of the Preisach model is a valueof the input voltage (control signal) received by the actuator. However,in other embodiments the input x of the Preisach model is a differentappropriate parameter.

In the above embodiments, the processor generates the control signal,i.e. the input x for the Preisach model, using the determined output y.However, in other embodiments the control signal is generated usingdifferent means, or a combination of means. For example, in otherembodiments the control is determined from a user input.

In the above embodiments, a wave-front sensor is used to provide thesignal corresponding to the detected feedback beam, i.e. a signalcorresponding to the curvature of the mirror. However, in otherembodiments a different appropriate device is used. For example, astrain gauge or capacitive sensor directly coupled to the mirror couldbe used.

In the above embodiments, hysteresis is compensated for in a deformablemirror of an adaptive optics system. However, in other embodiments,hysteresis is compensated for in any appropriate material of theadaptive optics system. Also, in other embodiments, hysteresis iscompensated for in other materials of other appropriate systems, forexample systems other than optics systems. In these embodiments, theinput and output of the Preisach model are different appropriateparameters.

1. A method of compensating for hysteresis in a system, the methodcomprising: determining a first estimate of a system parameter using thePreisach model; measuring a value of a system parameter; and determiningan updated estimate of the estimated system parameter using the measuredvalue of a system parameter.
 2. A method of compensating for hysteresisin a system according to claim 1, wherein: the measured system parameteris the same system parameter as the system parameter for which the firstestimate is determined using the Preisach model; and determining theupdated estimate of the estimated system parameter further uses thefirst estimate of the estimated system parameter.
 3. A method ofcompensating for hysteresis in a system according to claim 1, wherein:the measured system parameter is a further system parameter that is adifferent system parameter to the system parameter for which the firstestimate is determined using the Preisach model; and determining theupdated estimate of the estimated system parameter further uses adetermined value of the further system parameter, the determined valueof the further system parameter being determined in correspondence tothe first estimate of the estimated system parameter.
 4. A method ofcompensating for hysteresis in a system according to claim 1, wherein:the estimated system parameter is an input of the system; anddetermining a first estimate of the system parameter comprisescalculating the formula:$x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}$where x is the estimate of the input of the system; and x_(γ) _(i) _(δ)_(j) is the input of the system resulting from increasing an output ofthe system from a minimum output to a value of γ_(i) and then decreasingit to a value of δ_(j).
 5. A method of compensating for hysteresis in asystem according to claim 4, wherein calculating the formula$x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}$comprises: determining a value of x_(γδ) for each γ-δ pair in a discreteset of γ-δ pairs; and for all required values of x_(γδ) where γ and δare not in the discrete set, determining a value of x_(γδ) using aprocess of interpolation using values of x_(γδ) where γ and δ are in thediscrete set.
 6. A method of compensating for hysteresis in a systemaccording to claim 5, wherein the step of determining the updatedestimate of a system parameter comprises: if γ_(n) and δ_(n) are in thediscrete set, adjusting the value of x_(γ) _(n) _(δ) _(n) by an amountequal to a constant multiplied by the difference between the firstestimate of a system parameter and the measured value of a systemparameter; and if γ_(n) and δ_(n) are not in the discrete set, adjustingthe values of x_(γδ) for each γ-δ pair surrounding (γ_(n), δ_(n)) by anamount equal to a constant multiplied by the difference between thefirst estimate of a system parameter and the measured value of a systemparameter.
 7. A method of compensating for hysteresis in a systemaccording to claim 6, wherein the constant is equal to 0.005.
 8. Amethod of compensating for hysteresis in a system according to claim 5,wherein the step of determining the updated estimate of a systemparameter comprises: if γ_(n) and δ_(n) are in the discrete set,adjusting the value of x_(γ) _(n) _(δ) _(n) by an amount equal to aconstant multiplied by the difference between the determined value ofthe further system parameter and the measured value of a systemparameter; and if γ_(n) and δ_(n) are not in the discrete set, adjustingthe values of x_(γδ) for each γ-δ pair surrounding (γ_(n), δ_(n)) by anamount equal to a constant multiplied by the difference between thedetermined value of the further system parameter and the measured valueof a system parameter.
 9. A method of compensating for hysteresis in asystem according to claim 1, wherein: the estimated system parameter isan output of the system; and determining a first estimate of the systemparameter comprises calculating the formula:$y = {\sum\limits_{k = 1}^{n}\left( {y_{\alpha_{k}\beta_{k}} - y_{\alpha_{k}\beta_{k - 1}}} \right)}$where y is the estimate of the output of the system; and y_(α) _(i) _(β)_(j) is the output of the system resulting from increasing the input ofthe system from the minimum input to a value of α_(i) and thendecreasing it to a value of β_(j).
 10. A method of compensating forhysteresis in a system according to claim 9, wherein calculating theformula$y = {\sum\limits_{k = 1}^{n}\left( {y_{\alpha_{k}\beta_{k}} - y_{\alpha_{k}\beta_{k - 1}}} \right)}$comprises: determining a value of γ_(αβ) for each α-β pair in a discreteset of α-β pairs; and for all required values of y_(αβ) where α and βare not in the discrete set, determining a value of y_(αβ) using aprocess of interpolation using values of y_(αβ) where α and β are in thediscrete set.
 11. A method of compensating for hysteresis in a systemaccording to claim 10, wherein the step of determining the updatedestimate of a system parameter comprises: if α_(n) and β_(n) are in thediscrete set, adjusting the value of y_(α) _(n) _(β) _(n) by an amountequal to a constant multiplied by the difference between the firstestimate of a system parameter and the measured value of a systemparameter; and if α_(n) and β_(n) are not in the discrete set, adjustingthe values of y_(αβ) for each α-β pair surrounding (α_(n), β_(n)) by anamount equal to a constant multiplied by the difference between thefirst estimate of a system parameter and the measured value of a systemparameter.
 12. A method of compensating for hysteresis in a systemaccording to claim 1, wherein the system is an adaptive optics system.13. Apparatus adapted to perform the method of claim
 1. 14. A computerprogram or plurality of computer programs arranged such that whenexecuted by a computer system it/they cause the computer system tooperate in accordance with the method of claim
 1. 15. A machine readablestorage medium storing a computer program or at least one of theplurality of computer programs according to claim 14.